Embedded newform 177.2.f.a.2.9

• Label: 177.2.f.a.2.9
• Level: $177$
• Weight: $2$
• Character: 177.2
• Analytic conductor: $1.413$
• Analytic rank: $0$
• Dimension: $504$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	177.f (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.41335211578\)
Analytic rank:	\(0\)
Dimension:	\(504\)
Relative dimension:	\(18\) over \(\Q(\zeta_{58})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{58}]$

Embedding invariants

Embedding label			2.9
Character	\(\chi\)	\(=\)	177.2
Dual form			177.2.f.a.89.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.00955626 + 0.176255i) q^{2} +(-1.25381 - 1.19497i) q^{3} +(1.95730 + 0.212869i) q^{4} +(-1.32719 + 3.93896i) q^{5} +(0.222601 - 0.209571i) q^{6} +(0.403437 + 0.595025i) q^{7} +(-0.113337 + 0.691326i) q^{8} +(0.144093 + 2.99654i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 504 q - 27 q^{3} - 70 q^{4} - 29 q^{6} - 58 q^{7} - 19 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/177\mathbb{Z}\right)^\times\).

\(n\)	\(61\)	\(119\)
\(\chi(n)\)	\(e\left(\frac{1}{58}\right)\)	\(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.00955626	+	0.176255i	−0.00675729	+	0.124631i	0.993208	+	0.116349i	\(0.0371191\pi\)
							−0.999966	+	0.00828209i	\(0.997364\pi\)
\(3\)	−1.25381	−	1.19497i	−0.723889	−	0.689916i
							\(5\)	−1.32719	+	3.93896i	−0.593537	+	1.76156i	0.0529123	+	0.998599i	\(0.483150\pi\)
−0.646450	+	0.762957i	\(0.723747\pi\)
\(7\)	0.403437	+	0.595025i	0.152485	+	0.224898i	0.896257	−	0.443535i	\(-0.146276\pi\)
							−0.743772	+	0.668433i	\(0.766965\pi\)
\(11\)	3.62276	−	0.797430i	1.09230	−	0.240434i	0.367938	−	0.929850i	\(-0.380064\pi\)
							0.724365	+	0.689416i	\(0.242133\pi\)
\(13\)	1.33985	−	1.13808i	0.371607	−	0.315645i	−0.442051	−	0.896990i	\(-0.645749\pi\)
							0.813658	+	0.581344i	\(0.197473\pi\)
\(17\)	−2.22517	−	1.50870i	−0.539683	−	0.365914i	0.260684	−	0.965424i	\(-0.416052\pi\)
							−0.800367	+	0.599510i	\(0.795362\pi\)
\(19\)	−2.97560	+	5.61257i	−0.682649	+	1.28761i	0.263744	+	0.964593i	\(0.415043\pi\)
							−0.946392	+	0.323019i	\(0.895302\pi\)
\(23\)	4.18494	−	3.96419i	0.872621	−	0.826591i	−0.113341	−	0.993556i	\(-0.536155\pi\)
							0.985963	+	0.166965i	\(0.0533968\pi\)
\(29\)	−1.98792	+	0.107782i	−0.369147	+	0.0200146i	−0.237779	−	0.971319i	\(-0.576420\pi\)
							−0.131368	+	0.991334i	\(0.541937\pi\)
\(31\)	−2.43025	+	1.28844i	−0.436487	+	0.231410i	−0.672136	−	0.740428i	\(-0.734623\pi\)
							0.235650	+	0.971838i	\(0.424278\pi\)
\(37\)	−4.01719	+	0.658584i	−0.660422	+	0.108271i	−0.482667	−	0.875804i	\(-0.660332\pi\)
							−0.177755	+	0.984075i	\(0.556883\pi\)
\(41\)	0.489202	−	0.516444i	0.0764004	−	0.0806550i	−0.686674	−	0.726965i	\(-0.740930\pi\)
							0.763075	+	0.646310i	\(0.223689\pi\)
\(43\)	2.18178	−	9.91195i	0.332719	−	1.51156i	−0.452176	−	0.891929i	\(-0.649352\pi\)
							0.784895	−	0.619629i	\(-0.212717\pi\)
\(47\)	6.07516	−	2.04696i	0.886154	−	0.298580i	0.160825	−	0.986983i	\(-0.448584\pi\)
							0.725329	+	0.688403i	\(0.241688\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.f.a.2.9	✓	504
3.2	odd	2	inner	177.2.f.a.2.10	yes	504
59.30	odd	58	inner	177.2.f.a.89.10	yes	504
177.89	even	58	inner	177.2.f.a.89.9	yes	504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.f.a.2.9	✓	504	1.1	even	1	trivial
177.2.f.a.2.10	yes	504	3.2	odd	2	inner
177.2.f.a.89.9	yes	504	177.89	even	58	inner
177.2.f.a.89.10	yes	504	59.30	odd	58	inner